Proof. We first assume that \(\mu _j\) , \(j=0,\ldots ,n\) ,
are distinct and nonvanishing on \(\widetilde{\widetilde{\Omega }}\)  and
\(\widetilde{F}_{r}(\mu _j) \ne 0\)  on \(\widetilde{\widetilde{\Omega }}\) ,
\(j=0,\ldots ,n,\)  where \(\widetilde{\widetilde{\Omega }} \subseteq \Omega \) . Then,
the representation (REF ) for \(\phi \)  on \(\widetilde{\widetilde{\Omega }}\) 
follows by combining (REF ), (REF ), (), and
Theorem A.26 [1]}. The representation () for \(u\)  on
\(\widetilde{\widetilde{\Omega }}\)  follows from the trace formulas (REF )
and (F.59) [1]}. The representation () for \(\rho \)  on
\(\widetilde{\widetilde{\Omega }}\)  is clear from (REF ) and (REF ).
In fact, since the proofs of (REF ), (), and ()
are identical to the corresponding stationary results
in Theorem REF , which can be extended line by line to the
time-dependent setting, here we omit the corresponding details.
By continuity, (REF ), (), and () extend from
\(\widetilde{\widetilde{\Omega }}\)  to \(\Omega \) .
The
constraint (REF ) then holds on \(\widetilde{\widetilde{\Omega }}\)  by
combining (REF )-() and (F.58) [1]}. Equations
(REF ) and () are clear from (REF ) and ().
Again by continuity, (REF )-() extend from
\(\widetilde{\widetilde{\Omega }}\)  to
\(\widetilde{\Omega }\) .    \(\square \)
